Last Thoughts

By Nicolae Sfetcu

By Henri Poincaré - Translated by Nicolae Sfetcu

Henri Poincaré is a mathematician, physicist, philosopher and engineer, born April 29, 1854 in Nancy and died July 17, 1912 in Paris. He has carried out works of major importance in optics and in infinitesimal calculus. His advances on the problem of the three bodies make him a founder of the qualitative study of systems of differential equations and chaos theory; he is also a major precursor of the theory of special relativity and the theory of dynamical systems. Henri Poincaré is considered one of the last great universal scholars, mastering all branches of mathematics of his time and some branches of physics.
This book gathers here various articles and lectures that Henri Poincaré himself intended to form the fourth volume of his works of philosophy of science. All the previous ones had already appeared in this collection. It would be useless to recall their prodigious success. The most illustrious of modern mathematicians has been an eminent philosopher, one of those whose books profoundly influence human thought. It is probable that if Henri Poincaré himself had published this volume, he would have modified certain details, removed some repetitions. But it seemed to us that the respect due to the memory of this great death forbade any editing of his text.

• Language: English
• Publisher: Nicolae Sfetcu
• Release date: Sep 1, 2019
• ISBN: 9786060332916

Nicolae Sfetcu

Owner and manager with MultiMedia SRL and MultiMedia Publishing House. Project Coordinator for European Teleworking Development Romania (ETD) Member of Rotary Club Bucuresti Atheneum Cofounder and ex-president of the Mehedinti Branch of Romanian Association for Electronic Industry and Software Initiator, cofounder and president of Romanian Association for Telework and Teleactivities Member of Internet Society Initiator, cofounder and ex-president of Romanian Teleworking Society Cofounder and ex-president of the Mehedinti Branch of the General Association of Engineers in Romania Physicist engineer - Bachelor of Science (Physics, Major Nuclear Physics). Master of Philosophy.

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Last Thoughts

Henri Poincaré

Translator: Nicolae Sfetcu

Published by: MultiMedia Publishing

Copyright 2019 Nicolae Sfetcu

Published by MultiMedia Publishing, https://www.telework.ro/en/publishing/

ISBN: 978-606-033-291-6

Author: Henri Poincaré (1854 – 1912)

Original title: Dernières pensées (1913)

Publishing house: Ernest Flammarion

Translation: Nicolae Sfetcu

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Henri Poincaré is a mathematician, physicist, philosopher and engineer, born April 29, 1854 in Nancy and died July 17, 1912 in Paris. He has carried out works of major importance in optics and in infinitesimal calculus. His advances on the problem of the three bodies make him a founder of the qualitative study of systems of differential equations and chaos theory; he is also a major precursor of the theory of special relativity and the theory of dynamical systems. Henri Poincaré is considered one of the last great universal scholars, mastering all branches of mathematics of his time and some branches of physics.

This book gathers here various articles and lectures that Henri Poincaré himself intended to form the fourth volume of his works of philosophy of science. All the previous ones had already appeared in this collection. It would be useless to recall their prodigious success. The most illustrious of modern mathematicians has been an eminent philosopher, one of those whose books profoundly influence human thought. It is probable that if Henri Poincaré himself had published this volume, he would have modified certain details, removed some repetitions. But it seemed to us that the respect due to the memory of this great death forbade any editing of his text.

1. THE EVOLUTION OF LAWS

Mr. Boutroux, in his work on the contingency of the laws of nature, wondered if natural laws are not likely to change, if while the world is changing continuously, the laws themselves, that is to say to say the rules according to which this evolution takes place, will be the only ones exempt from any variation. Such a conception has no chance of ever being adopted by scholars; in the sense that they would understand it, they could not adhere to it without denying the legitimacy and the very possibility of science. But the philosopher retains the right to ask himself the question, to consider the various solutions it entails, to examine the consequences of it, and to seek to reconcile them with the legitimate demands of scientists. I would like to consider some of the aspects that the problem may take; I shall thus be led not to conclusions proper, but to various reflections which may not be without interest. If, on the way, I allow myself to speak a little longer about certain related questions, please forgive myself.

I

Let's first look at the mathematician's point of view. Let us admit for a moment that the physical laws have undergone variations in the course of ages, and ask ourselves if we should have a means of perceiving them. Let us not forget at first that the few centuries during which humanity lived and thought were preceded by incomparably longer periods in which man did not yet live; they will undoubtedly be followed by other periods when our species will have disappeared. If one wants to believe in an evolution of the laws, it cannot be, without contradiction, other than very slow, so that, during the few years that one thought, the laws of the nature could undergo only insignificant changes. If they have evolved in the past, then we must understand the geological past. Were the laws of former times those of today, the laws of tomorrow will still be the same? When we ask such a question, what meaning should we attach to words once, today and tomorrow? Today are the times whose history has preserved the memory; in the past, millions of years preceded history, and ichthyosaurs lived quietly without philosophizing; tomorrow, it is the millions of years that will come next, where the Earth will be cooled and where the man will have no more eyes to see or brain to think.

That put, what is a law? It is a constant link between the antecedent and the consequent, between the current state of the world and its immediate later state. Knowing the present state of every part of the universe, the ideal scientist who knew all the laws of nature would have fixed rules to deduce from them the state that these same parties will have the next day; it is conceivable that this process can be continued indefinitely. From Monday's state of the world, one will deduce that of Tuesday; knowing that of Tuesday, it will be deduced by the same methods that of Wednesday; And so on. But that's not all ; if there is a constant link between the state of Monday and that of Tuesday, we can deduce the second from the first, but we can do the opposite, that is to say that if we know the state of Tuesday, we can conclude that of Monday; from the state of Monday we will conclude likewise with that of Sunday, and so on; we can go back in time and we can go down. With the present and the laws, one can guess the future, but one can also guess the past. The process is essentially reversible.

Since we place ourselves here from the point of view of the mathematician, it is advisable to give this conception all the precision which it implies if we use mathematical language for it. We shall then say that the set of laws is equivalent to a system of differential equations which link the velocities of variations of the various elements of the universe to the present values ​​of these elements.

Such a system comprises, as we know, an infinity of solutions; but if we give ourselves the initial values of all the elements, that is to say their values at the moment t = 0, (the one that in the ordinary language we call the present) the solution is entirely determined, so that we can compute the values of all the elements at any time, that we suppose t > 0, which corresponds to the future, that we suppose t < 0, which corresponds to the past. What is important to remember is that the way of concluding from present to past is not different from how to conclude from the present to the future.

What means do we have then to know the geological past, that is to say the history of the times when the laws could have formerly varied? This past could not be directly observed and we only know it by the traces it has left in the present, we only know it by the present, and we can only deduce it from the process that we have just described, and which would also allow us to deduce the future. But is this process able to reveal changes in the laws? Obviously not; we cannot precisely apply it except by supposing that the laws have not changed; we know only the state of Monday, for example, and the rules which bind the state of Sunday to that of Monday; the application of these rules will then make us know the state of the Sunday; but if we wish to go further and deduce the state of Saturday, we must of necessity admit that the same rules which enabled us to go back up from Monday to Sunday, were still valid between Sunday and Saturday. Without this, the only conclusion we would be allowed is that it is impossible to know what happened on Saturday. If then the immutability of the laws appears in the premises of all our reasonings, we cannot to not find it in our conclusions.

Leverrier, knowing the current orbits of the planets, calculates, using Newton's law, what will have become these orbits in 10,000 years. However, he directs his calculations, he will never be able to find that Newton's law will be false in a few thousand years. He could, by simply changing the sign of time in his formulas, calculate what these orbits were 10,000 years ago; but it is certain in advance not to find that Newton's law has not always been true.

In short, we can know nothing of the past except on the condition of admitting that the laws have not changed; if we admit it, the question of the evolution of the laws does not arise; if we do not admit it, the question is insoluble, as are all those that relate to the past.

II

But it will be said, could it not be that the application of the preceding process would lead to a contradiction, or, if one wishes, that our differential equations admit no solution? Since the hypothesis of the immutability of laws, posited at the beginning of all our reasoning, would lead to an absurd consequence, we would have shown by absurdum that they have evolved, while being forever powerless to know in what sense.

Since our process is reversible, what we have just said applies in the future, and there seem to be cases where we could say that before that date the world must perish or change its laws; if for example the calculation shows us that at this date, one of the quantities we have to consider must become infinite, or take a physically impossible value. To perish, or to change its laws, is almost the same thing; a world that no longer has the laws of ours, it would no longer be our world, it would be another.

Is it possible that the study of the present world and its laws will lead us to formulas exposed to similar contradictions? Laws are obtained by experience; if they teach us that the state A of Sunday causes the state B of Monday, it is because we observed the two states A and B; therefore, none of these two states is physically impossible. If we continue the process, and if we conclude by passing every day from one day to the next, from state A to state B, then from state B to state C, then from state C to state D, etc., that is all these states are physically possible; because if the state D for example was not, one could never have done experiment proving that the state C generates after one day the state D. However far the deductions are pushed, we will never reach a physically impossible state, that is, a contradiction. If one of our formulas was not exempt, it would be that we would have exceeded the experience, it would have been extrapolated. Suppose, for example, that it has been observed that in such and such a circumstance the temperature of a body drops by one degree per day; if it is currently 20° for example, we will conclude that in 300 days it will be -280°; and this will be absurd, physically impossible, since the absolute zero is at -273°. What to say? Had it been observed that the temperature passed in a day from -279° to -280°? No, no doubt, since these two temperatures are unobservable. It had been seen, for example, that the law was true at about 0° and 20°, and it had been wrongly concluded that it should be true even to -273° and even beyond; an illegitimate extrapolation was made. But there is an infinity of ways to extrapolate an empirical formula, and among them one can always choose one that excludes physically impossible states.

We know the laws only imperfectly; experience only limits our choice, and among all the laws it allows us to choose, we will always find those which do not expose us to a contradiction of the kind of those which we have just spoken of and which could oblige us to conclude against immutability. This means of demonstrating such a development still escapes us, whether it is to demonstrate that the laws will change, or that they have changed.

III

At this point, we can oppose an argument of fact. You say that in seeking to ascend, thanks to the knowledge of the laws, from the present to the past, one will never meet with a contradiction, and yet the scientists have met with it, from which it does not seem that one can escape as easily as you think. That they are only apparent, that we may preserve the hope of lifting them, I grant you; but according to your reasoning, even an apparent contradiction should be impossible.

Let's quote an example immediately. If we calculate from the laws of thermodynamics, the time from which the sun has been able to send us its heat, we find about 50 million years; this time cannot suffice for geologists; not only has the evolution of organized forms not been able to take place so rapidly - this is a point on which we could discuss - but the deposition of layers where we find remains of plants or animals which do not were able to live without sun, required a number of years perhaps ten times greater.

What has made the contradiction possible is that the reasoning on which the geological evidence is based differs very much from that of the mathematician. Observing identical effects, we conclude to the identity of the causes, and for example by retracing the fossil remains of animals belonging to a family currently alive, we conclude that at the time when the layer containing these fossils, the conditions without which the animals of this family could not live, were all realized at once.

At first sight, it is the same thing that the mathematician did, whose point